Rapporto di Ricerca CS Simonetta Balsamo, Andrea Marin. On representing multiclass M/M/k queue by Generalized Stochastic Petri Net

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1 UNIVERSITÀ CA FOSCARI DI VENEZIA Dpartmento d Informatca Techncal Report Seres n Computer Scence Rapporto d Rcerca CS Febbrao 2007 Smonetta Balsamo, Andrea Marn On representng multclass M/M/k queue by Generalzed Stochastc Petr Net Dpartmento d Informatca, Unverstà Ca Foscar d Veneza Va Torno 55, 3072 Mestre Veneza, Italy

2 On representng multclass M/M/k queue by Generalzed Stochastc Petr Net Smonetta Balsamo, Andrea Marn Dpartmento d Informatca Unverstà Ca Foscar d Veneza Va Torno 55, 3072 Veneza Mestre, Italy {balsamo,marn}@ds.unve.t Abstract. In ths paper we study the relatons between mult-class BCMP-lke servce statons and generalzed stochastc Petr nets (GSPN). Representng queung dscplne wth GSPN models s not easy. We focus on representng mult-class queung systems wth dfferent queung dscplnes by defnng approprate fnte GSPN models. Note that queung dscplne n general affect performance measures n mult-class systems. For example, BCMP-lke servce centers wth Frst Come Frst Served (FCFS) and wth Last Come Frst Served wth Preemptve Resume (LCFSPr) have a (dfferent) product-form soluton under dfferent hypotheses. We defne structurally fnte GSPNs equvalent to the multclass M/M/k queung system wth FCFS, LCFSPR, Processor Sharng (PS) and Infnte Servers (IS). Equvalence holds n terms of steady state probablty functon and average performance measure. The man dea s to defne a fnte GSPN model that smulates the behavor of a gven queue dscplne wth some approprate random choce. Moreover, we prove that the combnaton of the ntroduced equvalent models has a closed-form steady state probablty by the M = M property. We consder queung systems wth both a sngle server wth load dependent servce rate, and multple servers wth constant servce rate. Introducton Queung theory and (Generalzed) Stochastc Petr Nets are mportant classes of stochastc models used to evaluate system performances. Queung systems have been wdely appled to represent resource contenton systems where a set of customers competes for resource usage. However, basc queung systems cannot model the synchronzaton between concurrent actvtes. Stochastc Petr nets (SPN) can be naturally used to represent systems wth synchronzaton and concurrency and to perform both qualtatve and quanttatve analyss. An mportant problem n system performance evaluaton based on performance models s the effcency of the soluton algorthms,.e. the ablty to defne classes of models that can be analyzed by methods wth a lmted space and tme computatonal complexty. Many results have been proposed n lterature whch gve effcent solutons of some types of stochastc models under certan condtons.

3 . INTRODUCTION Sngle queung system models have been wdely analyzed by consderng varous arrval dstrbutons, servce tme dstrbutons, classes of users and schedulng dscplnes. The sngle server queue s analyzed [4, [0, [3, [9 and several results have been derved for specal cases. Queung networks (QN) extend and combne varous queung systems to represent more complex systems. QN models can be represented by defnng an assocated stochastc process that, under some exponental and ndependence assumptons, s a contnuous-tme Markov Chan (CTMC) process. Although the statonary state probablty soluton of the assocated Markov Process, under stablty condtons, can be easly defned, the computaton can soon become unfeasble due to the hgh computatonal complexty. However, under certan assumptons QNs can be effcently analyzed by applyng the product-form theorem [5, whch defnes the steady state probablty functon as product of functons of each sngle servce center state. Product-form QNs can be analyzed by effcent algorthms (e.g. [8, [8, [7, [6) that yeld a low polynomal computatonal cost. SPNs, whch are defned n terms of a set of places and a set of exponentally tmed transtons connected by arcs, and a markng, whch s the state of the net, are represented by a CTMC process whose state space s the set of all possble markngs of the net. Specfcally the computaton of performance measures s neffcent because t requres to calculate the reachablty set, whch depends on the ntal markng and whose sze grows exponentally wth the number of places of the net and the number of tokens n the ntal markng. Henderson et al. ntroduced the dea of product-form for SPN n [2 and [ smplfyng the computaton of the performance ndexes. However, the algorthms for product-form SPNs stll requre to check condtons on the reachablty set. The class of Generalzed Stochastc Petr Nets (GSPN) allows modellng nets wth both exponentally tmed and mmedate transtons ntroducng more flexblty. The underlyng process of a GSPN can be defned as a sem-markov process. Product-form for GSPN has been studed by Balbo et al. n [3 and t s based on technques to reduce the problem to the product-form SPN theorem. Investgatng the relatons between classes of queung models and GSPN models s an nterestng problem and t has been consdered by some recent research lterature ([2, [20, [4). Most of these works focus on the relatons between QNs and SPNs or GSPNs and derve some equvalence results. However a lttle attenton has been devoted to the problem of representng varous types of schedulng dscplnes of queung systems by GSPNs. To the best of our knowledge, representng schedulng dscplnes n multclass models wth fnte GSPNs s stll an open problem. In [20 the authors ntroduce a comparson between QN models and SPN models based on the representaton of multclass features by colored Petr nets. However the dfferences between dfferent schedulng dscplnes are not analyzed. Balbo et al. n [2 combne GSPN and product-form QN by replacng subsystem n a low-level model wth ther flow equvalents models. Stll lttle attenton s devoted to schedulng dscplnes. In [4 the authors observe how they can map each servce staton of a BCMP QN to a complex GSPN. The GSPN model depends on the schedulng dscplnes but t has an nfnte 2

4 . INTRODUCTION number of places and transtons for the FCFS and LCFSPR statons. Then they gve a fnte and remarkably compact representaton by a GSPN equvalent to the detaled model. The compact representaton holds the product-form condtons for GSPN showed n [3 but t does not dstngush dfferent queung dscplnes by mappng everythng n the PS dscplne. Thus t s not possble to defne on the GSPN equvalent condtons to the ones dependng on the servce center schedulng dscplne of the BCMP theorem. On the other hand the detaled representatons of queung dscplne yeld non product-form GSPN models equvalent to BCMP QN. In ths paper we present an equvalence result between two types of stochastc models. We propose a fnte GSPN representaton of a set of queung systems wth varous schedulng dscplnes. Accordng to the BCMP-type servce centers we analyze Frst Come Frst Served (FCFS), Last Come Frst Served wth preemptve resume (LCFSPR), Processor Sharng (PS) and Infnte Servers (IS) schedulng dscplnes. The man dea behnd these results s a probablstc model of the queue,.e., all the customers of the same class wat n the same place and when a server becomes free the customer whch gets the servce s chosen n a probablstc way smlarly to what happens wth the random queung dscplne. In the LCFSPR dscplne, we also choose probablstcally the customer that looses the server when a new customer arrves to the system. The advantage of havng a fnte representaton whch s dfferent for the varous schedulng dscplnes s twofold: frst t makes the analyss easer. Second t does not requre the defnton of new semantc for the GSPN accordng to the queung dscplnes. Thus exstng analyss or smulaton tools can be used wth the GSPN nets defned n ths work. The proposed results are nterestng because they allow the representaton of an M/M/k queue wth varous queung dscplnes by a compact GSPN, whch s equvalent to the queung system n term of steady state queue length dstrbuton. A practcal consequence can be that t can extend a GSPN smulator or analyzer for analyzng multclass queue systems. The only requrement s that the tool s able to model state-dependent frng rates for tmed transtons and state-dependent weghts for mmedate transtons. There s no need to support the colored model extenson to represent dfferent classes. One could also ntegrate a GSPN analyzer by a functonalty that dentfes net structures equvalent to dfferent schedulng dscplnes M/M/k queung systems and then t can apply the closed form steady state formula. For the FCFS and PS dscplnes, the GSPN structure complexty,.e. the number of places and transctons, grows lnearly just wth the number R of classes of users, for the LCFSPR t grows lke O(R 2 ). We gve a GSPN model for the queung dscplnes consdered n the BCMP theorem [5. An open problem and a possble further research s the analyss of the GSPN models obtaned by combnng these blocks. Possbly, under some assumptons, t s possble to defne equvalence between the GSPN steady state probablty functon and the BCMP queung network steady state probablty functon. 3

5 2. GENERALIZED STOCHASTIC PETRI NETS The paper s structured as follows. Secton 2 brefly revews the GSPN models recallng formalsm we chose, Secton 3 revews some results of the queung systems theory used later n the paper. In Sectons 4, 5 we ntroduce the GSPNs respectvely equvalent to the FCFS and LCFS multclass M/M/k queue. Secton 6 dscuss the GSPN models for both PS schedulng and IS systems. The proof of some theorems are gven n appendx. Fnally, Secton 8 provdes some concludng remarks. 2 Generalzed Stochastc Petr Nets In ths secton we brefly recall the Generalzed Stochastc Petr Nets (GSPN). We consder the notaton for GSPN ntroduced n [5. In order to allow markng dependent probabltes for solvng conflcts among mmedate transtons we use the technques dscussed n [9. Then n the next secton we shall present GSPN models equvalent to queung systems under varous assumptons. Let us defne a marked Stochastc Petr Net whch conssts of a 8-tuple as follows: where: GSP N = (P, T, I(, ), O(, ), H(, ), Π( ), w(, ), m 0 ) P = {P,..., P M } s the set of M places, T = {t,..., t N } s the set of N transtons (both mmedate and tmed), I(t, p j ) : T P N s the nput functon, N, j M, O(t, p j ) : T P N s the output functon, N, j M, H(t, p j ) : T P N s the nhbton functon, N, j M, Π(t ) : T N s a functon that specfes the prorty of transton t, N, m N M denotes a markng or state of the net, where m represents the number of tokens n place P, N, w(t, m) : T N M R s the functon whch specfes for each tmed transton t and each markng m a state dependent frng rate, and for mmedate transtons a state dependent weght, m 0 N M represents the ntal state of the GSPN,.e. the number of tokens n each place at the ntal state. We consder ordnary nets,.e., functons I, O and H take values n {0, }. For each transton t let us defne the nput vector I(t ), the output vector O(t ) and the nhbton vector H(t ) as follows: I(t ) = (,..., M ) where j = I(t, P j ), O(t ) = (o,..., o M ) where o j = O(t, P j ) and H(t ) = (h,..., h M ) where h j = H(t, P j ). Functon Π(t ) assocates a prorty to transton t. If Π(t ) = 0 then t s a tmed transton,.e., t fres after an exponentally dstrbuted frng tme wth mean /w(t, m), where m s the markng of the net. If Π(t ) > 0 then t s an mmedate transton and ts frng tme s zero. We say that transton t a s enabled by markng m f m I(t a, p ) and m < H(t a, p ) for each =,..., M and no other transton of hgher prorty s enabled. We consder just 4

6 2. GENERALIZED STOCHASTIC PETRI NETS two prorty levels, 0 and. Hence when an mmedate transton s enabled all the tmed ones are dsabled. The frng of transton t changes the state of the net from m to m I(t )+O(t ). The reachablty set RS(m 0 ) of the net s defned as the set of all markngs that can be reached n zero or more frngs from m 0. We say that markng m s tangble f t enables only tmed transtons and t s vanshng otherwse. For a vanshng markng m let T α be the set of enabled mmedate transtons. Then the frng probablty for any transton t T α and any state m s denoted by p(t, m) and t s defned as follows: p(t, m) = w(t, m) t j T α w(t j, m). () Gven a tangble markng m the transton wth the lowest assocated stochastc tme fres. A GSPN s represented by a graph wth the followng conventons: tmed transtons are whte flled boxes, mmedate transtons are black flled boxes, places are crcles, f I(t, p j ) > 0 we draw an arrow from p j to t labelled wth I(t, p j ), f O(t, p j ) > 0 we draw an arrow from t to p j labelled wth O(t, p j ), f H(t, p j ) > 0 we draw an crcle endng lne from p j to t labelled wth the value of H(t, p j ), the markng m s represented by a set of m j flled crcles representng the tokens n place p j for each j =,..., M. For ordnary nets we do not use labels for the arrows. GSPN analyss conssts n fndng the steady state probablty for each tangble markng of the reachablty set. Some analyss technques are presented n [5. Under general assumptons, the stochastc process generated by the dynamc behavor of a standard SPN s a CTMC process. Mean state sojourn tmes are computed from the mean transton delays of the net. For GSPNs the dstrbuton of the sojourn tme n any markng can be expressed as a negatve exponental and determnstcally zero dstrbutons for tangble and vanshng markngs, respectvely. Thus the markng process can be studed as a sem-markov random process. The GSPN models ntroduced n ths paper present markng processes whch allow us to easly reduce the sem-markov process to a CTMC. In fact whenever a vanshng markng s reached, the next markng s tangble. Thus we can smply obtan a CTMC whose states are the tangble states of the orgnal process and the transton rates are computed weghtng the transtons rates of the orgnal process wth the frng probabltes of the mmedate transtons. Hence the mean sojourn tmes n the tangble states of the orgnal sem-markov process and the mean sojourn tmes of the CTMC are the same. Fnally let us ntroduce some other notatons: let e be an M-dmensonal vector wth all zero components but the -th whch s. We use the lower case t to name mmedate transtons, the upper case T to name tmed transtons, t to name a generc tmed or mmedate transton. 5

7 3. SINGLE QUEUING SYSTEMS WITH DIFFERENT CLASSES OF CUSTOMERS 3 Sngle Queung systems wth dfferent classes of customers In ths secton we brefly recall sngle queung systems wth dfferent classes of customers classfyng them on the number of servers and schedulng dscplnes. Let us consder an open queung system wth external arrvals, a queue, a set of dentcal servers and a set of R customer classes. The queung system s shown n Fgure. Customers of class r arrve at the system accordng to a Posson process wth rate λ r and requre an exponentally dstrbuted random servce tme wth parameter µ r, r =,..., R. The system has a set of ndependent servers, possbly nfnte. For sngle class queung systems some results n terms of steady state probablty hold for any schedulng dscplne that s work-conservatve and ndependent from the servce tme [0, [4. These results can be extended to multclass queung systems although they depend on the schedulng dscplne. We consder the followng dscplnes: Frst Come Frst Server (FCFS), Last Come Frst Server wth Preemptve Resume (LCFSPR), Processor Sharng (PS). The steady Server CLASS CLASS 2 Queue Server 2 CLASS R Server k Fg.. An M/M/k multclass queung system. state probablty of a M/M/k multclass system wth a specfc queung dscplne and constant servce rate s equvalent to the steady state probablty of a M/M/ multclass system wth the same queung dscplne and an approprate load-dependent servce rate. If all the customer servce tmes are dentcal,.e., µ r = µ for r =,..., R, the load dependent servce rate µ(j), where j s the number of customers at the system, s defned as follows: µ(j) = { jµ f j k kµ f j > k (2) 6

8 3. SINGLE QUEUING SYSTEMS WITH DIFFERENT CLASSES OF CUSTOMERS If the stablty condton R λ r kµ < holds, then we can evaluate the statonary queue length dstrbuton of the multclass M/M/k system for any schedulng dscplne by the correspondng M/M/ load dependent system. Let π (n) denote the steady state probablty of the M/M/k system, wth n = (n... n R ),.e., the probablty of fndng n customers of class for =,..., R n the system. Then we can wrte: π (n) = π 0 R λ n ( P r n R )! n R n! µ(j), (3) where π 0 s the probablty of fndng the system empty. When mean servce rates for dfferent customer classes are not dentcal,.e., µ µ j for j for some couple, j, then the load dependent servce rate functon µ r (n), for any class r and state n, s defned as follows: µ r (n) = n r n mn(n, k)µ r, n = R n. (4) The followng steady state probablty holds for LCFSPR and PS queung dscplnes: P R R π (n) = π 0 n )! R R ( R n λ n ) n j n! µ mn(k, ). (5) The stablty condton s k : n > k[ R r= λ r µ r (n) <. The BCMP theorem [5 consders servce centers wth sngle servers and state dependent servce tme. For FCFS servce statons the servce tme can depend only on the total number of customers n the system. Let n = R r= n r and x(n) be an arbtrary postve functon of n, representng the servce rate when there are n customers at the servce center relatve to the servce rate when n =. Then the steady state probablty functon s: π (n) = π 0 n! R R n! ( λ n n n µ) x(). (6) For LCFSPR and PS systems, BCMP theorem consders another state dependent servce rate. Let y r (n r ) be an arbtrary postve functon of n r whch denotes the servce rate of class r customers at servce center relatve to the servce rate when there s one class r customer at servce center.e. µ r. Then the steady state probablty functon s: π (n) = π 0 n! R R n! λ n R r= [( ) nr n r. (7) µ r y r (a) Note that these varous forms of state dependent servce rates can be combned. For example the steady state probablty (5) can be obtaned combnng equatons (6) and (7) by settng x(n) = mn(n,k) n and y r (n r ) = n r. 7 a=

9 4. REPRESENTING M/M/K/FCFS QUEUE BY GSPN 4 Representng M/M/k/FCFS queue by GSPN In ths secton we defne a GSPN that represents an R-multclass M/M/k/FCFS queue. Then we prove that the GSPN model s equvalent to the queung system n terms of the steady state probablty. Gven the M/M/k/FCFS models defned as n Secton 3 let us defne the model called GSPN-. Defnton (GSPN-). Accordng to GSPN defnton gven n Secton 2: P = P q P s {P 2R+ } wth P q = {P,..., P R } and P s = {P R+,..., P 2R }, T = T w T q where T q = {t,..., t R } and T w = {T R+,..., T 2R }, functon Π defned as follows: Π( t ) = { 0 f R + 2R f R, nput and output vectors for transton t, R: I(t ) = e + e 2R+ and O(t ) = e R+. Input and output vector for transton T R+ : I(T R+ ) = e R+ and O(T R+ ) = e 2R+, H(t ) = (0,..., 0) for all t T, w(t R+, m) = m R+ µ for R and w(t, m) = m for R, m 0 = (0,..., 0, k). Tokens arrve to places P, R accordng to Posson stochastc processes. Fgure 2 llustrates the graphcal representaton of GSPN- model where t,..., t R are mmedate transtons and T R+,..., T 2R are exponental transtons. Fg. 2. Graphcal representaton of model GSPN- Let m be a vald vanshng state of the GSPN-, and let T a T q be the set of mmedate transtons enabled by m, then the probablty of frng of t T a 8

10 4. REPRESENTING M/M/K/FCFS QUEUE BY GSPN can be wrtten as: p(t, m) = p (m) = m j {j t j T a} m j (8) We shall now derve a closed form soluton for the steady state probablty of GSPN- model by consderng the set of reachable markngs m = (m,..., m 2R+ ). Ths s gven by Lemma. Then we ntroduce a state aggregaton by defnng the aggregate state n = (n,..., n R ) where n = m + m R+, R. Ths state corresponds to the number of customers of class n the queung model. Theorem provdes the closed form soluton for model GSPN- n terms of aggregated statonary probablty of state n. Fnally the GSPN- model s shown to be equvalent to the M/M/k FCFS multclass queung system n terms of statonary probablty. Lemma. Let m = (m,..., m 2R+ ) be a reachable tangble state of the GSPN-. Then f the stablty condton holds, the statonary state probablty can be wrtten as follows: π(m) = π 0 R λ m +m R+ ( 2R =R+ m )! 2R =R+ m! ( P R m 2R )! m R m! µ(j). (9) where π 0 s a normalzng constant and µ(j) s the functon defned by (2). The proof s gven n appendx A and s based on verfyng the set of the CTMC global balance equatons. Theorem. Consder model GSPN- and let n = m + m R+, R and n = (n..., n R ) be an aggregated state. Let π a (n) be the steady state probablty of n for =,..., R. Then we can wrte: π a (n) = π 0 ( R n )! R n! r λ n P R n µ() n N R. (0) Proof. In order to derve equaton (0) we prove that: π a (n) = m m +m R+ =n R π(m), () for n N R and m n the reachablty set of model GSPN-. Consder the two followng cases: case ) R n k and case 2) R n < k. Case : R n k. Consder any combnaton of j wth r and 0 j n. Then the rght-hand sde of equaton () by usng formula (9) can 9

11 4. REPRESENTING M/M/K/FCFS QUEUE BY GSPN be wrtten as follows: j +...+j R =k =π 0 R j n π(n j, n 2 j 2,..., n R j R, j,..., j R, 0) =π 0 R =π 0 R λ n j j λ nj j λ n j j P R n P R n P R n µ(j) µ(j) µ(j) j +...+j r=k j n ( R n k)! R n k!! ( R n k)! R n k!! k! ( R n k)! R j R! (n j )! j +...+j R =k j n j +...+j R =k j n R n! R (n R j )! j! where the last sum satsfes the Vandermonde convoluton, thus we can wrte: π 0 R =π 0 R λ n j j λ n j j P R n P R n µ(j) µ(j) R ( n j ), ( R n ( k)! R R n k! n )! k ( R n )! n n,! whch s formula 0. Case 2: R n < k, that corresponds to the behavor of the queung system where all the customers are beng served and n GSPN- every place P wth R s empty. Note that n = m R+, so by equaton (9) we can wrte: π(0,..., 0, m R+,..., m 2R, l) = π 0 R λ n P R n µ() ( R n )! R n,! that yelds formula (0) and ths ends the proof. Corollary. The M/M/k queung system wth FCFS dscplne, R customer classes, arrval rates λ, R, sngle server rate µ and steady state probablty π (n) s equvalent to the GSPN- n terms of steady state probablty,.e., π a (n) = π (n) for all n N R where π a (n) s the aggregated probablty of GSPN gven by formula (0) Proof. It follows mmedately from equaton (3) and Theorem. Note that GSPN- model represents the M/M/k multclass system when the servce rate s ndependent from the class of the customers n servce. Thus t can not be used to represent LCFSPR or PS schedulng dscplnes. For example 0

12 5. REPRESENTING M/M/K/LCFSPR QUEUE BY GSPN consder the system wth LCFSPR queue, sngle server, and dfferent average servce rates for each class. Then we show now a counterexample to prove that the steady state gven by queung theory does not satsfy the GBE for the GSPN. Ths mples that GSPN- catches somehow the FCFS semantc (by not allowng preempton). Example. As example, consder a LCFSPR M/M/ queue wth two classes of customers wth average servce tme /µ and /µ 2. From queung theory we can wrte the steady state probablty as follows: π (n, n 2 ) = π (0, 0)λ n λn 2 2 (n + n 2 )! n!n 2! Suppose to represent ths system by GSPN- assocatng dfferent frng rates to transtons T 3 and T 4 : w(t 3, m) = m 3 µ and w(t 4, m) = m 4 µ 2. We calculate the effectve arrval rate to reachable tangble state m = (0, m 2,, 0, 0), wth m 2 > 0. The adjacent states are m = (0, m 2,, 0, 0), m 2 = (, m 2,, 0, 0), m 3 = (, m 2, 0,, 0), thus the effectve arrval rate to state m s: [ m 2 π(m) µ 2 λ 2 + λ (m 2 + ) µ λ 2 m 2 µ m λ 2(m 2 + ) µ 2 µ 2 m 2 + = π(m) [µ 2 + λ + λ 2. The effectve leavng rate for state m s clearly π(m)(µ + λ + λ 2 ), so the GBE on state m s satsfed f µ = µ 2. Fnally note that GSPN- can as well smulate a sngle server FCFS servce staton wth an BCMP-lke load dependent servce rate. We can state the followng lemma: Lemma 2. Let m = 2R m, x(m) be an arbtrary postve functon, m 0 = e 2R+ and let the frng rate of transton T R+,..., T 2R be w(t R+r ) = x(m)µ, r R. Then f m s a tangble reachable markng the steady state probablty functon s: π(m) = π 0 R λ m+m R+ ( R m )! R m! µ n µ n2 2 ( µ) P P 2R 2R m m. x(j). (2) The proof s gven n appendx. By defnng n = m +m R+ we can aggregate the states and we can prove that the steady state probablty π a of the aggregated CTMC s dentcal to π defned by equaton (6) of BCMP theorem. The net structure complexty s lnear on R, the number of customer classes. 5 Representng M/M/k/LCFSPR queue by GSPN In ths secton we ntroduce a GSPN whch can be consdered equvalent, for steady state probablty, to a multclass M/M/k queue wth LCFS wth preemptve resume schedulng dscplne. As we consder just exponentally dstrbuted

13 5. REPRESENTING M/M/K/LCFSPR QUEUE BY GSPN servce tmes, we do not consder the problem of representng the resume. We provde a model for ths queue system whose structure s fnte and depends only on the number of classes of customers,.e., not on the number of servers. A trval soluton could be obtaned by recallng that the steady state formula for a LCFSPR queue s equal to the Processor Sharng one so that we could use the same GSPN representaton. On the other hand we want to provde a model whch semantcally smulate closer the LCFS queue. Defnton 2 (GSPN-2). Accordng to GSPN defnton gven n Secton 2: P = P q P w P a {P 3R+ } where P q = {P,..., P R } and P w = {P R+,..., P 2R } and P a = {P 2R+,..., P 3R }, T = T q T w T f T g where T q = {t,..., t R } and T w = {T R+,..., T 2R } and T f = {t 2R+,..., t 3R } and T g = {t j,, j R}, functon Π s defned as follows: { f t T Π( t) = q T f T g, 0 f t T w Let, j R. The nput and output vectors of t T q : I(t ) = e + e 3R+ and O(t ) = e R+. The nput and output vectors for T R+ T w : I(t R+ ) = e R+ and O(t R+ ) = e 3R+. The nput and output vectors for t 2R+ T f : I(t 2R+ ) = e 2R+ + e 3R+ and O(t 2R+ ) = e R+. The nput and output vectors for t j T g : I(t j ) = e 2R+ + e R+j and e j + e R+, H(t ) = (0,..., 0) for t T q T w T f and H(t j ) = e 3R+ for t j T g, for, j R let w(t R+, m) = m R+ µ, w(t, m) = m, w(t 2R+, m) = and w(t j, m) = m R+j, m 0 = (0,..., 0, k). Tokens arrve to places P 2R+, R, accordng to Posson stochastc processes. Fgure 3 shows a graphcal model for R = 2 classes LCFSPR queue where dotted lnes are ntroduce for the sake of readablty and they do not have ant partcular meanng. Note that when a token arrves to the place P 2R+ t s temporally (.e. the state s vanshng) stored n P 2R+ and we have two cases: there s at least one free server,.e. m 3R+ > 0, thus the customer goes mmedately n servce. Ths s modelled by the mmedate transton set T f all the servers are busy,.e. m 3R+ = 0, so a customer s preempted and put n queue and the new customer goes n servce. Ths s modelled by R 2 transtons, T g. The nhbtor arcs are needed to avod pre-empton when there s at least one free server. Lemma 3. Consder the sets of mmedate transtons T q, T f, T g. Any two transtons belongng to two dfferent sets cannot be smultaneously enabled. Moreover any two transtons of T f cannot be enabled smultaneously, and f t a T g s enabled then just transtons t b T g wth b R can be enabled. 2

14 5. REPRESENTING M/M/K/LCFSPR QUEUE BY GSPN Fg. 3. Graphcal representaton of model GSPN-2. The proof mmedately derves by GSPN-2 structure. As consequence to lemma 3 we can solve the conflcts on mmedate transtons wth just one smple functon. When one or more transtons of T q are enabled, the probablty of frng for the -th transton s: m p(t, m) = p (m) = R l= m. (3) l When one or more transtons of T g are enabled, the probablty of frng s: m R+j p(t j, m) = p j (m) = R l= m. (4) R+l Now we can state a man lemma for model GSPN-2 representaton: Lemma 4. Let m = (m,..., m 3R+ ) be a reachable tangble markng of GSPN- 2 model. Then f the stablty condton holds, the statonary state probablty can be wrtten as follows: π(m) = π 0 R λ m +m R+ ( R m )! R m! ( R m R+)! R m R+! R ( ) m +m R+ µ P 2R m mn(j, k). (5) where µ s the average servce rate for one customer of class when there are no other customers n the system, k s the number of servers, π 0 s a normalzng constant. 3

15 6. REPRESENTING M/M/K/PS QUEUE AND M/M/ QUEUE BY GSPN The proof s gven n appendx A. Theorem 2. Consder model GSPN-2 and let n = m + m R+, R and n = (n,..., n R ) be an aggregated state. Let π a (n) be the steady state probablty of n for =,..., R. Then we can wrte: π a (n) = π 0 ( R n )! R n! R λ n R P R ( ) n n µ mn(k, ) n N R. (6) Proof. The proof s based on the Vandermonde formula and t s smlar the one gven for Theorem. Corollary 2. The M/M/k queung system wth LCFSPR dscplne, R customer classes, arrval rates λ, sngle server rate µ for class customers and steady state probablty π (n) s equvalent to model GSPN-2 n terms of steady state probablty,.e., π a (n) = π (n) for all n N R, where π a (n) s the aggregated probablty of GSPN gven by formula (6). The normalzng constant s π 0 = π(0,..., 0, k) = π (0,..., 0, k). Proof. It follows mmedately from queung theory and Theorem 2. The net GSPN-2 can as well smulate a sngle server LCFSPR servce staton wth a BCMP-lke load dependent servce rate. We can state the followng lemma: Lemma 5. Let m r = m r + m R+r, y r (m r) an arbtrary postve functon, m 0 = e 3R+ and let the frng rate of transtons T R+,..., T 2R be w(t R+r ) = y r (m r). The f m s a reachable tangble markng, the steady state probablty functon s: π(m) = π 0 R λ m+m R+ ( R m )! R m! R r= [( ) m mr+m r +m R+r R+r µ r a=. (7) y r (a) The proof s gven n appendx. By defnng n = m + m R+ we can aggregate the states and we can prove that the steady state probablty π a of the aggregated CTMC s dentcal to probablty π defned by equaton (7). For what concern the net structure complexty, the number of places grows as O(R) and the number of transtons grows as O(R 2 ). 6 Representng M/M/k/PS queue and M/M/ queue by GSPN The processor sharng dscplne can be easly represented consderng that the k processors are shared among the users n the system. Dfferent classes of users can have dfferent average tme servces, but all modelled by exponentally dstrbuted random varables. We can thnk that the k servers are shared among the R classes n proporton to the number of customers of the classes. 4

16 6. REPRESENTING M/M/K/PS QUEUE AND M/M/ QUEUE BY GSPN Defnton 3 (GSPN-3). Let us defne the model GSPN-3 as follows: P = {P,..., P R }, T = {T,..., T R }, Π(T ) = for each T T, I(T ) = e and O(T ) = (0,..., 0) for each T T, H(T ) = (0,..., 0) for each T T, w(t, m) = m m mn(k, m) where m = R m j for each T T, m 0 = (0,..., 0). Fgure 4 shows a graphcal representaton of the GSPN-3 model. Note that ths Posson Arrvals Fg. 4. Graphcal representaton of model GSPN-3 model s equvalent to a queung system wth PS dscplne and one server wth load-dependent exponental servce tme to smulate the mult-server feature. Therefore t mmedately follows the theorem: Theorem 3. Consder model GSPN-3. Then f stablty condton holds the statonary state probablty can be wrtten as follows: π(m) = π 0 ( R m )! R m! R λ m R P R ( ) m m µ mn(k, ), where µ s the average servce rate for one customer of class when there are no other customers n the system, k s the number of servers, π 0 s a normalzng constant. Ths model s smlar to the compact model ntroduced n [4, the only dfference s that we allow a whole state dependent frng rate thus we don t need a place whose tokens represent the total number of customers n the system. Model GSPN-3 can easly represent also the IS center. It suffces to set the frng rates of each transton T as m µ, R. 5

17 7. M M PROPERTY ON THE GSPN REPRESENTATION 7 M M property on the GSPN representaton Markov mples Markov property s ntroduced and studed by Muntz [7. In that paper he shows that f a queung system wth Posson arrvals presents departures accordng to a Posson process (M M property) then a combnaton of servce centers of ths type n a queung network has a product-form soluton. As we are consderng GSPNs we wll prove that a combnaton of GSPN-, GSPN-2 and GSPN-3 models stll holds a closed-form steady state probablty by defnng approprate traffc processes over the CTMC assocated to each of the models and usng the results gven n [6 whch generalze Muntz s work. We now brefly revew Melamed s results lmted to a CTMC n steady state. Consder an ergodc CTMC wth state space Γ and a set of traffc transtons denoted by Θ,..., Θ R, where Θ Γ Γ, Θ. Let us defne K (t) as the process whch counts the number of transtons (α, β) Θ up to t. Let m = γ Γ η Θ π(η)ξ(η γ) and for each state γ Γ let m (,γ) (γ) = η Θ π(η)ξ(η γ) where Θ (,γ) (, γ) = {β (β, γ) Θ } and ξ(η γ) s the transton rate between states η and γ. Then we can state that K (t) are mutually ndependent Posson processes f and only f the followng equaton holds: γ Γ, R R m (γ) = π(γ) m (8) We am to study the departure traffc processes from our models. Take for example model GSPN-, we can defne R traffc processes as follows: Θ = {(m, m) : m = m + }, =,..., R, (9) where m = m + m R+. In our case, n order to prove that K (t) are ndependent Posson processes when there are Posson arrvals, t suffces to prove that: γ Γ, π(η)ξ(η γ) = λ π(γ), (20) η Θ (,γ) In appendx we prove that ths condton holds for GSPN-, GSPN-2 and GSPN- 3 models by defnng approprate traffc processes. As observed n [6 ths property of the CTMC s equvalent to the M = M gven by Muntz thus t assures that a BCMP-lke composton of these GSPN models holds a closed-form steady state probablty functon. Random swtches between the blocks and user class swtches can be easly modelled by mmedate transtons. 8 Fnal remarks In ths paper we have shown how to represent mult-class sngle queung systems by structurally fnte GSPN for varous queung dscplnes. For each of the BCMP center types we have ntroduced a GSPN model whose steady state 6

18 A. APPENDIX probablty, aggregatng on the number of customers n the system for each class, s equal to the correspondent sngle queue servce center. Hence the two models are equvalent n terms of steady state dstrbuton and average performance ndexes. The man advantages of our representaton are the followng. We defne a fnte GSPN model. The abstracton level of the GSPN model allows the representaton of the queung behavor wthout ntroducng a hgh level of detals n the state specfcaton. We dstngush the customers watng n the queue from those beng served wthout takng n account the arrval order. Ths allows, as well as a fnte representaton, a steady state probablty whch s less detaled than the proposed n [4 whch consders the sngle staton detaled representaton wth the order of the customers n the queue, smlarly to the BCMP paper [5. On the other sde the models we propose are more detaled than those whch just consder the total number of customers n a center as the compact models of [4. The FCFS and the LCFSPR (or PS) schedulng dscplnes have dfferent GSPN representatons and the FCFS can not be used to represent the other ones f the servce depends on the customer class. The GSPN models smulate the correspondng queung system even f ther semantc s dfferent. The man dea of the defnton of the GSPN models s the way we represent the customer of the queung system whch gets the free server and the customer whch looses a server n case of preempton. In both cases we model the customer choce of a class wth a random selecton, accordng to the probablty proportonal to the number of customers n queue (or beng served) of that class over the total number of customers n queue (number of servers). The M M property allows us to state that a combnaton of GSPN-, GSPN- 2 and GSPN-3 models smlar to the servce centers combnaton n BCMP networks, has a smple closed form steady state probablty. In [ authors defne a queung center somorphc to GSPN- and show how t can be embedded n a BCMP queung network so that the steady state probablty functon of the network does not change. In the GSPN formalsm probablstc routng can be easly smulated by ntroducng a block wth a place and an mmedate transton for each possble route just after the tmed transtons of the models. Further research deals wth the extenson of the proposed LCFSPR model to Coxan servce tme dstrbutons and the defnton of algorthms to dentfy GSPN whch are compostons of models GSPN-, GSPN-2, GSPN-3 and n order to obtan effcently a set of sgnfcant performace ndexes. A Appendx Hereafter, for the sake of smplcty, by referrng to the relatons between the queueng system and GSPN models, we refer to and mx the notaton of queung theory and Petr nets. For example, we can say watng customers to refer to tokens n place P,.., P R of GSPN- or GSPN-2, customers beng served to refer to the tokens n P R+,..., P 2R of GSPN- or GSPN-2, customer arrval to refer 7

19 A. APPENDIX to token arrval, free servers to refer to the presence of tokens n the place P 2R+ or P 3R+ n the GSPN- or GSPN-2 respectvely. In our vew ths descrpton should make the followng proofs easer to understand. A. Proof of Lemma Proof. We consder four cases. Case ) Suppose that all servers are busy and at least one customer s watng, so m 2R+ = 0 and m > 0 for some =,..., R. Consder state m = (m,..., m R, m R+,..., m 2R, 0). Clearly we have that 2R =R+ m = k that s the number of servers. Consder the tangble markngs from whch m s reachable possbly through the frng of mmedate transtons combned wth tmed ones and the correspondng transton rates:: M α = {m = m e m > 0} wth rate λ M β = {m = m + e m R+ > 0} wth rate (m R+ )µp (m ); M γ = {m = m + e e R+ + e R+j m R+ > 0, j } wth rate (m R+j + )µp (m ). The set of markngs reachable from m and the correspondng transton rates can be classfed as follows: M a = {m = m + e } wth rate λ ; M b = {m = m e m R+ > 0, m > 0} wth rate (m R+ )µp (m); M c = {m = m e + e R+ e R+j m R+j > 0, m > 0, j} wth rate (m R+j )µp (m). We prove that equaton (9) satsfes the global balance equatons (GBE) by checkng that the effectve arrval rate to a state due to a new customer s equal to the effectve leavng rate due to a job completon, and the effectve arrval rate due to a job completon s equal to the effectve leavng due to new customer arrval: π(m )ξ(m, m) = π(m) ξ(m, m ) m M α m M b M c π(m )ξ(m, m) = π(m) ξ(m, m ) = m M β M γ m M a π(m )ξ(m, m) = π(m) ξ(m, m ) m M α M β M γ m M a M b M c where ξ(a, b) represents the transton rate from markng a to markng b. The effectve leavng rate from state m due to completon of a job s: [ π(m) m j+k >0 (m j+k )µ = π(m)(kµ). 8

20 A. APPENDIX The effectve arrval rate to state m due to arrvals to the system s: [ π(m )ξ(m, m) = π(m) ( m λ m M α X [ m = π(m)(kµ) R m = π(m)(kµ), j X 2R R m µ( j m j )λ ) where X = { m > 0, R}. Note that by hypothess 2R m k thus: 2R µ( m ) = kµ. Consder state m and set Y = {j m R+j > 0, j R}. The effectve leavng rate from state m due to arrvals of customers to the system s π(m) R λ. The effectve arrval rate to state m s: [ π(m) + j Y R Y j ( R λ m ) + j m j + λ j m R+ m R+j + m + ( R g= m g) + [ = π(m) j Y [ = π(m) j Y [ R = π(m) λ. µ(( 2R m ) + ) m m j + R+jµ R m + ( R g= m g) + m + µ(( 2R g= m g) + ) (m R+j + )µ λ j kµ m R+jµ + λ j m R+j k + R R Y j Y j λ j m R+ kµ µ m R+ λ j k Case 2) Consder now the case when all servers are busy, but places P,..., P R are empty. Consder a generc state m = (0,..., 0, m R+,..., m 2R, 0). The markngs from whch m s reachable are classfed as: M α = {m = m e R+ + e 2R+ m R+ > 0} wth rate λ ; M β = {m = m + e m R+ > 0} wth rate (m R+ )µp (m ); M γ = {m = m + e e R+ + e R+j m R+ > 0, j } wth rate (m R+j + )µp (m ). The markngs reachable from m and ther effectve rates are the same as n case ). Let Y be the set defned n case ). We now prove that the effectve 9

21 A. APPENDIX arrval rate to state m from markngs n M α s equal to the effectve leavng rate from m due to job completon whch s π(m)(kµ). [ π(m) ( m [ R+ µ(k)λ ) = π(m) m R+ µ = π(m)(kµ). λ k Y We prove now that the effectve arrval rate from the markngs n M β M γ s equal to the effectve leavng rate from m. [ π(m) (λ j µ(k + ) m R+jµ) + j Y [ =π(m) j Y λ j m R+j k + R Y j R Y j λ j Y m R+ λ j = π(m) k m R+ m R+j + µ(k + ) (m R+j + )µ [ R λ j. Case 3) Assume that there are not tokens n places P,..., P R and at least one server s free and at least one s busy, that s 0 < m 2R+ < k. Then the effectve leavng rate from m s smply π(m)[µ R m R+j + R λ j. The states from whch m s reachable and the correspondng rates are: M α = {m = m e R+ + e 2R+ m R+ > 0} wth rate λ ; M β = {m = m + e R+ e 2R+ } wth rate (m R+ + )µ. Let Y be defned as n case ). We now prove that the effectve arrval rate to state m s equal to the effectve leavng rate from m. [ π(m) + Y R Y λ [ =π(m) + [ =π(m) ( R m R+ R m R+j µ( λ R m R+j + m R+ + λ m R+ R m R+j R m R+j )λ µ( R m R+j + ) (m R+ + )µ R ( m R+j )µλ λ R m R+j + m R+ + R R m R+j )µ + λ. ( R m R+j + )µ (m R+ + )µ Case 4) Assume that the system s empty, that s m 2R+ = k. Ths case s trval. The effectve leavng rate from m s clearly π(m)[ R λ. The effectve arrval rate to m can just be due to a job completon and t s easy to show that s s equal to the leavng rate. 20

22 A. APPENDIX A.2 Proof of Lemma 2 Proof. The proof verfes that equaton (2) satsfes the set of GBE for the CTMC assocated to the net. We consder two cases. Case) Let m = (m,..., m R, 0,..., 0) + e R+ be a reachable tangble markng. State m can be reached from states m = m + e r+j e r+ + e due to a job completon wth rate x( + 2R r= m r)µp (m ). So the total effectve arrval rate due to a job completon s: [ R π(m) λ j + R r= m r m + x( + 2R r= m r)µ x( + 2R r= m + m r )µ + 2R r= m r R = π(m) λ j, whch s the total leavng rate from m due to arrvals to the system. State m can be reached from states m = m e j where j X = {j m j > 0, j R} wth rate λ j. So the effectve arrval rate due to an arrval to the system s: [ π(m) λ j j X m j 2R R k= m x( k k= [ 2R m k )µλ j = π(m) x( k= m k )µ, whch s dentcal to the total leavng rate due to a job completon. Case 2) If m 2R+ = the proof s trval. A.3 Proof of Lemma 4 Proof. The proof s smlar to the proof of Lemma. We prove that equaton (5) satsfes the GBE of the CTMC assocated to GSPN-2 by verfyng the partal balance. We consder agan four cases. Case ) Take a reachable markng m wth m 3R+ = 0 (.e. all servers are busy), and there s at least one token n one of the places P, wth R (.e. there s at least one customer n queue). Let X = { m > 0, R} and Y = { m R+ > 0, R}. We verfy that the effectve arrval rate to state m due to a job completon s equal to the effectve leavng rate from m due to a customer arrval and that the effectve arrval rate to state m due to a customer arrval s equal to the effectve leavng rate due to a job completon. State m s reachable due to an arrval event from states m = m e wth X Y wth rate λ p (m ) or from states m = m e + e R+ e R+j wth 2

23 A. APPENDIX X, j Y and j wth rate λ j p j (m ) Hence we can wrte: [ m π(m) R s= m µ kλ m R+ k s X Y λ + j Y [ =π(m) R s= m s [ =π(m) R s= m s X j λ j j X Y m m R+j R s= m s m R+ + µ m R+ + jkλ j k m j m R+j µ j + R s= m s µ j m R+j m j Y R [ R µ j m R+j m = π(m) µ j m R+j, X whch s dentcal to the effectve leavng rate from m due to a job completon. State m s reachable, due to job completon, from states m = m + e wth rate m R+ µ p (m ) and from states m = m + e e R+ + e R+j wth rate m R+j µ j p (m ). Hence the effectve arrval rate due to a job completon can be wrtten as: [ ( R π(m) λ m j) + m + m R+ µ m + kµ R m j + + Y X j R ( R k= λ m k) + m R+ m + j (m R+j + )µ j m + m R+j + kµ j ( R k= m k) + Y Y j [ m R+ =π(m) λ k + R Y j m R+ λ j = π(m) k [ R λ, whch s dentcal to the effectve leavng rate form state m due to a customer arrval to the system. Case 2) Consder now state m wth m 3R+ = 0 and m = 0 wth R (.e. no customers n queue). The leavng rates from m for a job completon or an arrval are the same as case. State m s reachable, due to a job completon, form states m = m + e e R+ + e R+j wth Y and j wth rate (m R+j + )µ j p j (m ) and from states m = m + e wth Y and rate m R+ µ p (m ). Hence the effectve arrval rate to m due to a job completon can be calculate as follows: [ π(m) λ m R+ µ + kµ Y [ =π(m) λ m R+ + k k Y R Y j m R+ (m R+j + )µ j = m R+j + kµ j R [ R m R+ λ j = π(m) λ j, Y j 22

24 A. APPENDIX whch s dentcal to the effectve leavng rate from m due to an arrval to the system. State m s reachable, due to an arrval to the system, from states m = m e + e 3R+ wth Y wth rate λ. Hence the effectve arrval rate to state m due to a customer arrval can be calculated as follows: [ π(m) Y m [ R+ [ R R s= m µ kλ = π(m) m R+ µ = π(m) m R+ µ, R+s λ whch s dentcal to the effectve leavng rate from m due to a job completon. Case 3) Consder the case of m wth m 3R+ = b < k. State m s reachable, due to an arrval, from states m = m e R+ + e 3R+ wth rate Y wth rate λ, thus the effectve arrval rate due to a customer arrval s: [ π(m) Y λ Y m [ R+ R e s= m µ (k b)λ = π(m) m R+ µ, R+s whch s dentcal to the effectve leavng rate from m due to a job completon. State m s reachable, due to a job completon, from states m = m+e R+ e 3R+ wth rate (m R+ + )µ, thus the effectve arrval rate due to a job completon s: [ R π(m) λ ( R s= m R+s m R+ + (m R+ + )µ µ k b + = π(m) R λ, whch s dentcal to the effectve leavng rate from m due to an arrval to the system. Case 4) consders m 3R+ = k,.e. when the system s empty, and t s trval. A.4 Proof of Lemma 5 Proof. The proof verfes that equaton (7) satsfes the set of GBE for the CTMC assocated to the net. We consder two cases. Case ) Take the tangble reachable markng m = (m,..., m R, 0,..., 0) + e R+. State m can be reached from states m = m + e r+j e r+ + e due to a job completon wth rate y j (m j + m R+j )µ j p (m ). So the effectve arrval rate to m due to a job completon s: [ R R k= π(m) λ m k + j m + m + j j (m j + )µ j y j (m j + )µ R j k= m k + = π(m) R λ j, whch s dentcal to the total leavng rate from m due to arrvals to the system. State m can be reached from state m = m + e r+j e j e R+k where j X = 23

25 A. APPENDIX {j m j > 0, j R} wth rate λ j. So the effectve arrval rate to m due to an arrval to the system s: [ m j π(m) λ R k= m y (m + m R+ )µ λ = π(m) [y (m + m R+ )µ, k j X whch s dentcal to the total leavng rate from m due to a job completon. Case 2) If m 3R+ = the proof s trval. A.5 Proof of Independence of the departure processes from GSPN models GSPN-: Proof. Consder model GSPN-. Let Θ = {(m, m) : m = m + } for =,..., R. In order to verfy equaton (20) consder a generc tangble marknf m. We consder the followng cases: ) m 2R+ = 0, and 2) m 2R+ > 0. Case ) Let m be a reachable tangble state wth m 2R+ = 0, then: Θ (, m) = {m m = m + e R+ + e j e R+j, m R+j > 0, j } {m m = m + e, m R+ > 0},, j R. Let sgn(m ) be the ndcator functon defned as follows: sgn(m ) = when m > 0 and 0 otherwse, and let Y = {j m R+j > 0, j R}. The left hand sde of equaton (20) can be rewrtten as follows: [ π(m) j Y j λ + R a= m a m j + m R+j m R+ + kµ (m R+ + )µ + R a= + sgn(m )λ m a m + kµ m m + R+µ + R a= m a ( ) = π(m) [λ = π(m)λ, j Y j m R+j k + m R+ k m j + + R a= m a whch gves the rght hand sde of equaton (20). Case 2) Le m be a reachable tangble state wth m 2R+ > 0. Then Θ (, m) = {m + e R+ e 2R+ }. Thus equaton (20) holds, n fact: + π(m) [λ R a= m R+a m R+ + ( + R a= m R+a)µ ( + m R+)µ = π(m)λ. Ths proves that the traffc processes assocated to Θ, R, are pontwse ndependent Posson processes,.e., the departure processes for each class of customers are Posson ndependent processes under ndependent Posson arrvals. 24

26 A. APPENDIX GSPN-2: Proof. Consder model GSPN-2. In ths model m = m +m R+, wth R and let Θ = {(m, m) : m = m +}. In order to verfy equaton (20) consde a generc tangble state m. We consder the followng cases: ) m 3R+ = 0, and 2) m 3R+ > 0. Case ) Let m be a reachable tangble state wth m 3R+ = 0, then: Θ (, m) = {m m = m + e R+ + e j e R+j, m R+j > 0, j } {m m = m + e m R+ > 0}, j R. Let sgn(m ) be the ndcator functon defned as follows: sgn(m ) = when m > 0 and sgn(m ) = 0 otherwse, and let Y = {j m R+j > 0, j R}. The left hand sde of equaton (20) can be rewrtten as follows: [ π(m) j Y j λ + R a= m a m j + + R a= + sgn(m )λ m a m + ( =π(m) [λ j Y j m R+j k + m R+ k m R+j m j + (m R+ + )µ m R+ + kµ + R m + m R+ µ kµ + R ) = π(m)λ, a= m a a= m a whch s the rght hand sde of equaton (20). Case 2) Le m be a reachable tangble state wth m 3R+ > 0. Then Θ (, m) = {m + e R+ e 3R+ }. Thus equaton (20) holds, n fact: + R a= π(m)[λ m a (m R+ + )µ = π(m)λ. m R+ + (m R+ + )µ GSPN-3 Proof. Consder model GSPN-3. Let Θ = {(m, m) m = m + e, R}. Provng that equaton (20) holds s trval. References. Afshar, P. V., Bruell, S. C., and Kan, R. Y. Modelng a new technque for accessng shared buses. In Proceedngs of the Computer Network Performance Symposum (New York, NY, USA, 982), ACM Press, pp Balbo, G., Bruell, S. C., and Ghanta, S. Combnng queueng network and generalzed stochastc Petr nets for the soluton of complex models of system behavor. IEEE Transactons on Computers 37 (998),

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